6 Ring extensions and equivalences

In this section we investigate the behaviour of equivalences as considered in the previous sections under ring extensions.

6.1 Ring extensions. Let α : R → A be a morphism of associative rings with units. Related to it we have the induction functor

R-Mod→A-Mod, M 7→A⊗RM, and therestriction functor

A-Mod→R-Mod, _AN 7→RN .

For a givenR-moduleP, puttingS = End_R(P) andT = End_A(A⊗RP) we have the ring morphism

β :S →T, f 7→id⊗f.

We will be interested inA-modules (resp.,T-modules) which have certain properties asR-modules (resp.,S-modules). Refining the notations introduced before we set

add^A_R(P) = {V ∈A-Mod |RV ∈add(P)}, Add^A_R(P) = {V ∈A-Mod |RV ∈Add(P)}, Gen^A_R(P) = {V ∈A-Mod |RV ∈Gen(P)}, Pres^A_R(P) = {V ∈A-Mod |RV ∈Pres(P)}, Stat^A_R(P) = {V ∈A-Mod |^RV ∈StatR(P)}, Adst^T_S(P) = {X ∈T-Mod |^SX ∈Adst(P)}. It is easy to see (e.g., [13, Lemma 1.2]) that

A⊗RP ∈Gen(P) if and only if Gen_A(A⊗RP) = Gen^A_R(P).

From Hom-tensor relations (e.g., [28, 15.6]) we have the 6.2 Basic isomorphisms.

(1) For any V ∈A-Mod there is a functorial S-module isomorphism ϕ : HomR(P, V)→HomA(A⊗^RP, V), f 7→[a⊗p7→a·(p)f].

(2) For V =A⊗^RP we get HomR(P, A⊗^RP)'T and theR-module isomorphism id⊗ϕ :P ⊗SHom_R(P, A⊗RP)→P ⊗ST.

Our main interest is to transfer properties of the R-module P to the A-module A⊗RP. It turns out that the conditions we are looking at can be transferred provided A⊗RP is P-static as an R-module. With the above preparations we can prove the following crucial result (see [20, Theorem 4.9, 5.5]):

6.3 Related equivalences. Assume A⊗RP ∈Stat^A_R(P). Then:

(1) Stat^A_R(P) = Stat_A(A⊗RP).

(2) Adst^T_S(P) = Adst(A⊗^RP) and there is an equivalence Hom_A(A⊗RP,−) : Stat^A_R(P)→Adst^T_S(P).

(3) If P is self-small then we have an equivalence

Hom_A(A⊗RP,−) : Add^A_R(P)→Add^T_S(S).

In each case the inverse functor is (A⊗RP)⊗T −.

Proof. A⊗RP ∈Stat^A_R(P) means P ⊗SHom_R(P, A⊗RP)'A⊗RP.

(1) Combined with an isomorphism from 6.2 we get the R-module isomorphism P ⊗ST 'A⊗RP.

This implies the isomorphisms for anyV ∈A-Mod,

(A⊗RP)⊗T Hom_A(A⊗RP, V) '(P ⊗ST)⊗T Hom_A(A⊗RP, V) 'P ⊗SHom_A(A⊗RP, V)

'P ⊗SHom_R(P, V).

Now V ∈ Stat^A_R(P) means by definition that the last expression in this chain is isomorphic to V, whereas V ∈ Stat_A(A ⊗RP) means that the first expression is isomorphic to V.

(2) First notice the R-module isomorphisms for any T-moduleV, P ⊗SV '(P ⊗ST)⊗T V '(A⊗RP)⊗T V.

Assume V ∈Adst^T_S(P). Then we have the isomorphisms

V 'Hom_R(P, P ⊗SV) 'Hom_R(P,(A⊗RP)⊗T V) 'Hom_A(A⊗RP,(A⊗RP)⊗T V), provingV ∈Adst_T(A⊗RP).

The same chain of isomorphisms shows Adst_T(A⊗RP)⊂Adst^T_S(P).

In view of (1) and the first part of (2) the final assertion about the equivalence follows from the basic equivalence 2.4 applied to A⊗RP.

(3) We know that Add^A_R(P)⊂Stat^A_R(P) = Stat_A(A⊗RP), and it is easy to verify that H_P(Add^A_R(P))⊂Add^T_S(S)⊂Adst^T_S(P). 2

Combining the preceding observations we obtain:

6.4 w-Σ-quasi-projective modules. Let P be a w-Σ-quasi-projective R-module and assume A⊗^RP ∈Pres(P). Then:

(1) A⊗RP is a w-Σ-quasi-projective A-module.

(2) If P is ^P-self-static we have an equivalence

Hom_A(A⊗RP,−) : Pres^A_R(P)→Adst^T_S(P), with inverse functor (A⊗^RP)⊗^T −.

(3) If P is a self-small R-module then A⊗RP is a self-small A-module.

Proof. (1) By [30, 3.2], factor modules of P-presented modules by P-generated modules are P-presented. This implies

Pres_A(A⊗RP)⊂Pres^A_R(P),

and by the functorial isomorphism in 6.2 we conclude that A⊗^R P is w-Σ-quasi-projective as anA-module.

(2) By 4.5, P-presented modules are P-static and so A⊗RP ∈ Pres(P) implies A⊗RP ∈Stat^A_R(P). Hence by 6.3, we have the inclusions

Pres_A(A⊗RP)⊂Pres^A_R(P) = Stat^A_R(P) = Stat_A(A⊗RP)⊂Pres_A(A⊗RP).

From this and 2.4 we obtain the equivalence as given.

(3) By [30, 5.1], forP self-small and w-Σ-quasi-projective, Hom_R(P,−) commutes with direct limits of P-presented modules. Since A⊗RP ∈ Pres(P), any infinite direct sum (A⊗RP)^(Λ) is the direct limit of its finite partial sums and hence

Hom_A(A⊗RP,(A⊗RP)^(Λ)) ' Hom_R(P,(A⊗RP)^(Λ)) ' Hom_R(P, A⊗RP)^(Λ)

' Hom_A(A⊗RP, A⊗RP)^(Λ).

2 6.5 Self-tilting modules. Let P be a self-tilting and ^P-self-static R-module, and assume A⊗RP ∈Gen(P). Then A⊗RP is a self-tilting A-module and we have an equivalence

HomA(A⊗^RP,−) : Gen^A_R(P)→Adst^T_S(P), with inverse functor (A⊗RP)⊗T −.

Proof. By 6.4, we have the equalities

Gen_A(A⊗RP) = Gen^A_R(P) = Pres^A_R(P) = Pres_A(A⊗RP) = Stat_A(A⊗RP).

2 Recall that P is a tilting module in R-Mod if it is self-tilting and a subgenerator inR-Mod. We consider some special cases of this.

6.6 Corollary. Assume A⊗RP ∈Gen(P). Then:

(1) IfP is a projective generator in σ[P], thenA⊗RP is a projective generator in σ_A[A⊗RP].

(2) IfP is self-tilting andGen(P)is closed under extensions inR-Mod, thenA⊗RP is self-tilting and Gen_A(A⊗RP) is closed under extensions in A-Mod.

(3) Assume that P is ^P-self-static and (i) A_R is flat or (ii) _AA is finitely cogen-erated andA⊗^RP is a faithful A-module.

If P is tilting in R-Mod, then A⊗^RP is tilting in A-Mod.

Proof. (1) Recall thatP is a projective generator inσ[P] if and only if it is self-tilting and a generator in σ[P].

Let P be a projective generator in σ[P]. Then P is ^P-self-static by 3.4, and by 6.5, A⊗^RP is self-tilting. Since σA[A⊗^RP] ⊂ Gen^A_R(P) = GenA(A⊗^RP) we see that A⊗RP is a generator in σ_A[A⊗RP].

(2) Suppose Gen(P) is closed under extensions inR-Mod. Since Gen_A(A⊗RP) = Gen^A_R(P) this implies that Gen_A(A⊗RP) is closed under extensions in A-Mod.

(3) Suppose R∈ σ[P], i.e., there is a monomorphism R →P^k, for some k ∈IN. IfA_R is flat, thenA'A⊗RR→A⊗RP^k is a monomorphism and henceA⊗RP is a subgenerator inA-Mod. If _AA is finitely cogenerated and A⊗RP is faithful, then A⊂(A⊗RP)^k, for some k ∈IN. Now the assertions follow from 6.5. 2 Remarks. Let P be finitely generated. IfP is self-tilting then it is a ∗-module and 6.5 implies Fuller [13, Theorem 2.2]. Condition (1) in 6.6 corresponds to [13, Corollary 2.4]. If P satisfies the condition in 6.6(2) then P is called quasi-tilting in R-Mod (see [8]) and we obtain [13, Corollary 2.5].

Acknowledgements. The author is very indebted to John Clark for helpful discussions on the subject.

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